Suppose you are a prairie dog assigned to guard duty with other
prairie dogs from your town. When you see a predator coming, you have
two choices: sound the alarm or remain silent. If you sound the
alarm, you help ensure the safety of the other prairie dogs, but you
also encourage the predator to come after you. For you, it is safer
to remain silent, but if all guards remain silent, everyone is less
safe, including you. What should you do when you see a predator?

This scenario is an example of the Volunteer’s Dilemma, a game similar
to the Prisoner’s Dilemma discussed in Section 10.5. In
the Prisoner’s Dilemma each player has two options—cooperate and
defect. In the Volunteer’s Dilemma each player also has two
options—volunteer (sound the alarm in our prairie dog example) or
ignore (remain silent). If one player volunteers then the other
player is better off ignoring. But if both players ignore, both pay a
high cost.

In the Prisoner’s Dilemma, both players are better off
if they both cooperate; however, since neither knows what the other
will do, each independently comes to the conclusion that he or she
should defect.

In the Volunteer’s Dilemma, it is not immediately
clear what outcome is best for both players. Suppose the players
are named Alice and Bob. If Alice volunteers, Bob is better off ignoring;
if Alice ignores, Bob is better off volunteering. This does not
provide a clear best strategy for Bob. By the same
analysis, Alice reaches the same
conclusion: if Bob volunteers, she is better
off ignoring; and if Bob ignores, she is better off volunteering.

Instead of one optimal outcome, as in the Prisoner’s Dilemma, there
are two equally good outcomes: Bob volunteers
and Alice ignores, or Bob ignores and Alice volunteers.

In the prairie dog town, there are more than two guards. As in the
scenario with just Alice and Bob, only one player needs to volunteer
to benefit the entire town. So there are as many good outcomes as
guards—in each case, one guard volunteers and the others ignore.

If one guard is always going to volunteer, though, then there is
little point in having multiple guards. To make the situation more
fair, we can allow the guards to distribute the burden of volunteering among
themselves by making decisions randomly. That is, each guard
chooses to ignore with some
probability γ or to volunteer with probability 1 − γ.
The optimal value for γ is that where each player volunteers as
little as necessary to produce the common good.

Marco Archetti investigates the optimal value of γ in his
paper “The Volunteer’s Dilemma and the Optimal Size of a Social
Group.” This section replicates his analysis.

When each player volunteers with the optimal probability,
the expected payoff of volunteering is the same as the expected payoff
of ignoring. Were the payoff of volunteering higher than the payoff
of ignoring, the player would volunteer more often and ignore less;
the opposite is true were the payoff of ignoring higher than the
payoff of volunteering.

The optimal
probability depends on the costs and benefits of each
option. In fact, the optimal γ for two individuals is

γ2 = c/a

where c is the cost of volunteering, and a is the total cost
if nobody volunteers. In
the prairie dog example, where the damage of nobody volunteering is
high, γ2 is small.

If you transfer more prairie dogs to guard duty, there are more
players to share the cost of volunteering, so we expect the
probability of each player ignoring should increase. Indeed, Archetti
shows:

γN = γ21/(N−1)

γN increases with
N, so as the number of players increases, each player volunteers
less.

But, surprisingly, adding more guards does not make the town safer.
If everyone volunteers at the optimal probability, the probability that
everyone except you ignores is γNN−1, which is γ2, so
it doesn’t depend on N. If you also ignore with probability γN,
the probability that everyone ignores is γ2 γN, which
increases with N.

This result is disheartening. We can ensure that the high-damage
situation never occurs by placing the entire burden of the common good
on one individual who must volunteer all the time. We could instead
be more fair and distribute the burden of volunteering among the
players by asking each of them to volunteer some percentage of the
time, but the high-damage situation will occur more frequently as the
number of players who share the burden increases.

Exercise 1

Read about the Bystander Effect at
http://en.wikipedia.org/wiki/Bystander_effect. What
explanation, if any, does the Volunteer’s Dilemma provide for the
Bystander Effect?

Exercise 2

Some colleges have an Honor Code that requires students to report
instances of cheating. If a student cheats on an exam, other students
who witness the infraction face a version of the volunteer’s
dilemma. As a witness, you have two options: report the offender or
ignore. If you report, you help maintain the integrity of the Honor
Code and the college’s culture of honesty and integrity, but you incur
costs, including strained relationships and emotional discomfort. If
someone else reports, you benefit without the stress and hassle. But
if everyone ignores, the integrity of the Honor Code is diminished, and
if cheaters are not punished, other students might be more likely to
cheat.

Download and run thinkcomplex.com/volunteersDilemma.py, which
contains a basic implementation of the Volunteer’s Dilemma. The code
plots the likelihood that nobody will volunteer with given values of
c and a across a range of values for N. Edit this code to
investigate the Honor Code volunteer’s dilemma. How does the
probability of nobody volunteering change as you modify the cost of
volunteering, the cost of nobody volunteering, and size of the
population?

At some colleges, cheating is commonplace and students seldom report
cheaters. At other colleges, cheating is rare and likely
to be reported and punished. The explicit and implicit rules about
cheating, and reporting cheaters, are social norms.

Social norms influence the behavior of individuals: for example, you
might be more likely to cheat if you think it is common and seldom
punished. But individuals also influence social norms: if more
people report cheaters, fewer people will cheat.
We can extend the analysis from the previous section to model these
effects.

Our model is based on a genetic algorithm presented by Robert Axelrod
in “Promoting Norms: An Evolutionary Approach to Norms.” Genetic
algorithms model the process of natural evolution. We create a
population of simulated individuals with different attributes. The
individuals interact in ways that test their fitness (by some
definition of “fitness”). Individuals with higher fitness are more
likely to reproduce, so over time the average fitness of the
population increases.

In the cheating scenario, the relevant attributes are:

boldness

: the likelihood that an individual cheats, and

vengefulness

: the likelihood that an individual reports
a cheater.

The players interact by playing two games, called
“cheat-or-not” and “punish-or-not.” In the first, each
player decides whether to cheat, depending on the value of
boldness. In the second, each player decides whether to
report a cheater, depending on vengefulness.
These subgames are played several times per generation so each
player has several opportunities to cheat and punish cheaters.

The fitness of each player depends on how they play the
subgames. When a player cheats, his
fitness increases by reward points, but the fitness of others
decreases by damage points. Each individual who reports a
cheater loses cost fitness points and causes the cheater to
lose punishment fitness points.

At the end of each generation, individuals with the highest fitness
levels have the most children in the next generation.
The properties of each individual in the new generation are then
mutated to maintain variation.

This code summarizes the structure of the simulation:

for generations in range(many):
for steps in range(repetitions):
for person in persons:
cheat_or_not()
punish_or_not()
genetic_repopulation()
genetic_mutation()

Each run starts with a random population and runs for 300 generations.
The parameters of the simulation are reward = 3, damage = 1, cost
= 2, punishment = 9. At the end, we compute the mean boldness and vengefulness of the population.
Figure 14.1 shows the results; each dot represents one
run of the simulation.

There are two clear clusters, one with low boldness and one
with high boldness. When boldness is low, the
average vengefulness is spread out, which suggests that
vengefulness does not matter very much. If few people cheat,
opportunities to punish are rare and the effect of vengefulness on
fitness is small. When boldness is high,
vengefulness is consistently low.

In the low-boldness cluster, average fitness is higher, so
individuals in the high-boldness cluster would be better off if they
could move. But if they make a unilateral move toward higher
boldness or higher vengefulness, their fitness suffers. So the
high-boldness scenario is stable, which is why this cluster exists.

Figure 14.2: Proportion of good outcomes, varying the cost of reporting cheaters and the punishment for cheating.

Suppose you are founding a new college and thinking about the academic
culture you want to create. You probably prefer an
environment where cheating is rare. But our simulations suggest that
there are two stable outcomes, with low and high rates of cheating.
What can you do to improve the chances of reaching (and staying in)
the low-cheating regime?

The parameters of the simulation affect the probability of the
outcomes. In the previous section, the parameters were reward = 3,
damage = 1, cost = 2, punishment = 9. The probability of
reaching the low-cheating regime was about 50%. If you were
founding a new college, you might not like those odds.

The parameters that have the strongest effect on the outcome are
cost, the cost of reporting a cheater, and punishment, the cost
of getting caught. Figure 14.2
shows how the proportion of good outcomes depends on these parameters.

There are two ways to increase the chances of a good outcome,
decreasing cost or increasing punishment. If we fix
punishment=9 and decrease cost to 1, the probability of a good
outcome is 80%. If we fix cost=2, to achieve the same probability,
we have to increase punishment to 11.

Which option is more appealing depends on practical and cultural
considerations. Of course, we have to be careful not to take this
model too seriously. It is a highly abstracted model of complex human
behavior. Nevertheless, it provides insight into the emergence
and stability of social norms.

Exercise 3

In this section we fixed reward and damage, and explored the
effect of cost and punishment.
Download our code from thinkcomplex.com/normsGame.py and
run it to replicate these results.

Then modify it to explore the effect of reward and damage. With
the other parameters fixed, what values of reward and damage give
a 50% chance of reaching a good outcome? What about 70% and 90%?

Exercise 4

The games in this case study are based on work in
Game Theory, which is a set of mathematical methods for analyzing the
behavior of agents who follow simple rules in response to economic
costs and benefits. You can read more about Game Theory at
http://en.wikipedia.org/wiki/Game_theory.